What exactly are Prestige classes, and what purpose do they serve? Are they, in any way, better than the default classes? Do they have any disadvantages/advantages to them?
Warlock Prestige Class Eligibility
Basically, there are three rules:
Spell-like abilities have caster levels, and therefore things that require a particular caster level can be met with a spell-like ability that has sufficient caster level. Warlock invocations are specifically arcane, which means they can also meet requirements for arcane caster level.
Having spell-like abilities is not the same as knowing how to cast spells; no number of spell-like abilities allows one to qualify as being “able to cast spells” of any level, even if one knows spell-like abilities that emulate spells of a higher level.
Spell-like abilities that mimic particular spells, on the other hand, do count as those spells for the purposes of requirements.
Complete Arcane explains each of these on page 71. Page 18 describes how a warlock benefits from a spellcasting prestige class.
What Prestige Classes Does a Warlock Actually Qualify For?
List questions are not on topic here, but the short answer is, “not many.” Wizards made a few in Complete Arcane, but both before and after Complete Arcane, being able to cast spells of a given level have been by-far the most common sort of requirement for a spellcasting prestige class. Complete Arcane does, at least, list each of its own prestige classes by which base classes each is best for, so you can search Table 2–1 on page 18 for that.
From other books, Complete Mage and Dragon Magic both add new invocations, and I know Complete Mage does have a few classes that warlocks can qualify for. There are other places scattered around that have prestige classes they can take, but there are not many.
I do recommend houseruling the warlock qualifications to allow them to take prestige classes that require casting spells of a given level.
There are simply too many prestige classes that are too fitting and appropriate to deny them to warlocks. Just allow them to qualify for a prestige class if a wizard of their (effective) level could cast spells of the required level. Some class benefits (bonus spell slots, metamagic effects, extra spells known) are meaningless for them, but still, a discerning player should be able to find useful things. The warlock is not a high-powered class; it’s a good idea to offer them easy upgrades like this.
For those particularly enamored of the RAW...
The notorious Precocious Apprentice feat from Complete Arcane can... arguably be taken by warlocks. If that is the case, they’d get a 2nd-level spell slot for doing so, and can then take prestige classes that require 2nd-level spells. Said prestige classes can then progress their warlock invocations as normal. This is not recommended; just houserule warlock prestige class entry and be done with it.
Notable Warlock Prestige Classes
Just a couple of prestige classes that deserve special mention.
Of course, the best warlock prestige class is the hellfire warlock from Fiendish Codex II: Tyrants of the Nine Hells. This class is three-levels long, adds +2d6 hellfire blast damage each level, and advances invocations on each level. Hellfire blasts also do Constitution damage to yourself, and the class specifically states that you cannot use hellfire blast if you are immune to Constitution damage. A one-level dip in binder from Tome of Magic for Naberius, the Grinning Hound is the most common way to heal that up quickly and easily. The strongheart vest soulmeld might work, depending on how your DM reads the immunity rule (strongheart vest basically gives you DR for ability damage, but it does not give immunity; YMMV)—strongheart vest, as well as the incarnate class or Shape Soulmeld and Open Chakra feats you need to access it, are from Magic of Incarnum.
Hellfire warlock is so good that a lot of warlock optimizers abuse legacy champion (Weapons of Legacy), which has the class feature of advancing another class’s features (like +1 spellcasting level, but for any sort of level and advancing anything). By RAW, this allows you to have more than three effective hellfire warlock levels, and therefore more than the +6d6 damage that it usually grants. A lot of DMs take a pretty dim view of that rule, however.
This prestige class from Races of Destiny is not a warlock prestige class, does not advance any warlock class features, and nothing about warlock makes it easier to enter (it only requires a race, a feat, and some skills). What it does do, however, is give you a “floating feat”—a 2nd-level chameleon’s bonus feat can be changed every day.
This is relevant to warlocks because at 12th level, they gain imbue item, the ability to craft magical items without knowing the requisite spell. They still need the feat, however: the second level of chameleon means that they can swap in whatever Item Creation feat they like for these purposes, and on days when they’re not crafting, they can just swap it to Extra Invocation. Along similar lines, they can swap it to Extra Invocation (the dead walk) to build an undead army on down days, then swap it to something combat-oriented when they go into battle – without the dead walk, they won’t be able to animate any more minions, but they will keep the minions they’ve already made.
There's No Limit to the Number of Prestige Classes a Character Can Take...
...That is, if the character qualifies for the prestige classes. A character who qualifies for multiple prestige classes can take levels in multiple prestige classes. In fact, it's often a good idea to do exactly that so you end up playing the kind of character you want. The DM can limit which prestige classes are available in his campaign, however, so asking first is always a good idea.
It's Just Powered, Not Overpowered
Characters get things from prestige classes, but as some base classes get almost nothing for staying in their base classes, that's not a big deal. Further, meeting requirements can be challenging for the more intense prestige classes, sucking up a lot of resources that can be put toward things a character might rather do or be good at, and classes that gain high-level class features (e.g. druid) will miss those when taking a prestige class. A prestige classes is a trade-off--often a very good trade-off but a trade-off nonetheless.
O, and just to be clear: I'm not saying there aren't crazy powerful prestige classes... because there totally are crazy powerful prestige classes. Instead, I'm saying the idea of taking multiple prestige classes isn't, in itself, unbalanced.